Plus construction

In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups. It was introduced by Kervaire (1969), and was used by Daniel Quillen to define algebraic K-theory. Given a perfect normal subgroup of the fundamental group of a connected CW complex $X$, attach two-cells along loops in $X$ whose images in the fundamental group generate the subgroup. This operation generally changes the homology of the space, but these changes can be reversed by the addition of three-cells.

The most common application of the plus construction is in algebraic K-theory. If $R$ is a unital ring, we denote by $GL_{n}(R)$ the group of invertible $n$-by-$n$ matrices with elements in $R$. $GL_{n}(R)$ embeds in $GL_{{n+1}}(R)$ by attaching a $1$ along the diagonal and ${\displaystyle 0}$s elsewhere. The direct limit of these groups via these maps is denoted $GL(R)$ and its classifying space is denoted $BGL(R)$. The plus construction may then be applied to the perfect normal subgroup $E(R)$ of $GL(R)=\pi _{1}(BGL(R))$, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For $n>0$, the $n$th homotopy group of the resulting space, $BGL(R)^{+}$ is the $n$th $K$-group of $R$, $K_{n}(R)$.

See also

• Semi-s-cobordism

References

• Adams (1978), Infinite loop spaces, Princeton, N.J.: Princeton University Press, pp. 82–95, ISBN 0-691-08206-5
• Kervaire, Michel A. (1969), "Smooth homology spheres and their fundamental groups", Transactions of the American Mathematical Society, 144: 67–72, doi:10.2307/1995269, ISSN 0002-9947, MR 0253347
• Quillen, Daniel (1971), "The Spectrum of an Equivariant Cohomology Ring: I", Annals of Mathematics. Second Series, 94 (3): 549–572, doi:10.2307/1970770.
• Quillen, Daniel (1971), "The Spectrum of an Equivariant Cohomology Ring: II", Annals of Mathematics. Second Series, 94 (3): 573–602, doi:10.2307/1970771.
• Quillen, Daniel (1972), "On the cohomology and K-theory of the general linear groups over a finite field", Annals of Mathematics. Second Series, 96 (3): 552–586, doi:10.2307/1970825.

External links

• Hazewinkel, Michiel, ed. (2001), "Plus-construction", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4